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Geo-Proofs
Quadrilaterals
Name: _________________________________
For #1-16, determine if each conditional is true or false. If it is true, write a 2-column proof to prove the
statement. If it is false, draw a counterexample and name the figure that you drew. Caution: read very,
very carefully.
1) If both pairs of opposite angles in a quadrilateral are congruent, then the quadrilateral is a parallelogram.
T
2) If two sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
F – draw an isosceles trapezoid
3) If a diagonal and the sides of quadrilateral WXYZ form two congruent triangles, then WXYZ is a
parallelogram.
F – draw a kite
4) If a pair of consecutive angles in a quadrilateral are congruent, then the quadrilateral is a parallelogram.
F – draw a rt. trapezoid
5) A quadrilateral is a parallelogram if it has two pairs of congruent sides.
F – draw a kite
6) A quadrilateral is a parallelogram if it has one pair of congruent sides and one pair of parallel sides.
F – draw an isosceles trapezoid
7) If one pair of opposite sides of a quadrilateral are both parallel and congruent, then the quadrilateral is a
parallelogram.
T
8) If two sides of a quadrilateral are perpendicular, then the quadrilateral is a rectangle.
F- draw a rt. trapezoid
9) If a quadrilateral has congruent diagonals, then it is a rectangle.
F – draw an isosceles trapezoid
10) If a quadrilateral has opposite sides congruent, then it is a rectangle.
F – draw a rhombus
11) If a quadrilateral has diagonals that bisect each other, then it is a rectangle.
F – draw a rhombus
12) If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
T
13) If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
T
14) If the diagonals of a quadrilateral are perpendicular, then it is a rhombus.
F – draw a kite
15) If a quadrilateral has all four sides congruent, then it is a square.
F – draw a rhombus
16) If a quadrilateral has both pairs of opposite sides congruent and one right angle, then it is a rectangle.
T
17) If one angle of a parallelogram is a right angle, then the parallelogram is a rectangle.
T
If possible, draw a trapezoid that has the following characteristics. If the trapezoid cannot be drawn,
explain why.
18) three congruent sides
Yes, draw an isosceles trapezoid where the 2 legs have the same measure as one of the bases.
19) a leg longer than either base
YES. draw a "tall" one
20) two right angles
YES. draw a rt. trapezoid
21) one pair of opposite angles are congruent
NO. Would lead you to a proof for a parallelogram (use AAS to show 
sides – then you have "one pair of opp. sides being ║ and  to make it a
and then use CPCTC for the other
)
22) congruent bases
NO. Since the bases are already parallel – we now have "one pair of opp. sides both ║ and " – so it would be a
and not a trapezoid.
23) bisecting diagonals
NO. If one pair of opp. sides have to be parallel, then the fact that the diagonals bisect each other would lead
you directly to a proof for it being a parallelogram. (show 
using SAS and use CPCTC to get the 3rd side,
showing that you have one pair of opp. sides both ║ and  - so it would be a )
24) four acute angles
NO. The sum of 4 angles each less than 90 would be a total of a sum less that 360. Since all quads have 360
degrees – and a trapezoid is a quad – then it's impossible for such a figure to be a trapezoid (or even a quad for
that matter).